I bought a 5 years old unknown fig tree potted in 20 gal large container in last Oct and I am thinking to proceed root pruning right now or not. I tried this afternoon, but it was too big and heavy to handle by myself. Therefore I only removed 40% of the potting soil and replaced with new Pro-Mix potting mix. Then I found the draining of the large container is not very good. so can I use drill to make some small holes from the bottom and lower part of the container to ensure better drainage without remove the root ball outside from the container ? The unknown fig tree is about to wake up from dormant now. I am not sure if this will kill the root ball not to grow well later on the year or not. 

I know this wasn't your question, but I'm having trouble understanding whether the roots needed pruned or not.  It sounds like you didn't remove the tree from the pot to actually see the condition of the roots.  If that's true, there's no way to tell if changing some of the soil will be good enough.  If the root ball is massive and circling heavily around the outside of the pot, the roots will need to be pruned or eventually the tree will decline/die.  Then again, if you truly did change 40% of the soil (I'm not sure how to do that without removing the tree from pot), the tree should be fine through the coming season.

As to your question, drilling into the pot (even with the root ball in there) won't hurt anything.  Remember, these are the roots you would chop off when root pruning anyway.  Once they make it to tree size, figs are amazingly hardy plants that can overcome quite a lot of mishandling and still thrive.

FWIW, if you will be setting the pot on dirt, you might want to consider drilling holes on the sides of the pot near the bottom.  That way, if roots grow out of the holes into the ground they'll be easier to cut off when you want to move your tree.  I can't take credit for the tip, but I believe it will save you lots of aggravation in the fall.

You can make holes for drainage using drill and it will help only if the medium is not compacted and water runs easily through it. It will not damage the plant at this stage. However, it is hardly substitute for root pruning after five years growth. 
If you are placing pot on the soil or partially burying it during summer growth time then holes at the bottom become problematic during fall when pot is to be moved for winter storage because roots passes through thee holes into the soil that becomes hart to cut. Holes on the sides near the bottom cause no such problem.

If you are going to place it on the concrete then there is no problem with holes on the bottom as well. 
I place my pots on the soil and cover the side holes with soil&mulch so the root can go out into the soil in search of moisture and nutrient instead of being confined to the soil contents inside the pot. I cut these outside roos in the fall to move the pots for winter storage. Roots emerging from the pot bottom take effort in our cold wet fall.